Spatiotemporal accessible solitons in fractional dimensions

We report solutions for solitons of the “accessible” type in globally nonlocal nonlinear media of fractional dimension (FD), viz., for self-trapped modes in the space of effective dimension $2<D\le 3$ with harmonic-oscillator potential whose strength is proportional to the total power of the wave field. The solutions are categorized by a combination of radial, orbital, and azimuthal quantum numbers $(n,l,m)$. They feature coaxial sets of vortical and necklace-shaped rings of different orders, and can be exactly written in terms of special functions that include Gegenbauer polynomials, associated Laguerre polynomials, and associated Legendre functions. The validity of these solutions is verified by direct simulations. The model can be realized in various physical settings emulated by FD spaces; in particular, it applies to excitons trapped in quantum wells.

Figure 1

FD solutions for $D=2.1$. (a) The structure of the fundamental solution with $n=l=m=0$, shown (here and below) in terms of Cartesian coordinates $x,y,\tau$ corresponding to the spherical coordinates. (b) Intensity ${\left|\mathrm{\Theta }\left(\theta \right)\right|}^2$ produced by Eq. (3b) for $l=0$ and $m=0$.

Figure 2

Multilayered FD solutions for $m=0$ and various sets of the two other quantum numbers: (a) $n=2, l=2$; (b) $n=1, l=3$; (c) $n=0, l=4$. The color is used here and in other figures only to distinguish different layers in the solutions.

Figure 3

Asymmetric necklace-ring solutions for various sets of quantum numbers $\left(n,l,m\right)$. (a) Horizontal ring with $l=m$ and different $n$: left $\left(n,l,m\right)=\left(0,7,7\right)$, center $\left(n,l,m\right)=\left(1,5,5\right)$, right $\left(n,l,m\right)=\left(2,3,3\right)$. (b) A vertical ring with $m=1$ and different $l$: left $\left(n,l,m\right)=\left(0,7,1\right)$, center $\left(n,l,m\right)=\left(1,5,1\right)$, right $\left(n,l,m\right)=\left(2,3,1\right)$. Recall that the color is used to distinguish different features in the intensity distribution, and is not related to the level of intensity.

Figure 4

Necklace solitons: (a) $n=0, l=8$, and $m=6,5,4$ from left to right, respectively; (b) $n=1, l=6$, and $m=4,3,2$ from left to right.

Figure 5

Comparison of exact solution (5) with the numerically found counterpart. (a) The analytical solution given by Eq. (5); (b) results of numerical simulations of Eq. (1) at $\zeta =80$ initiated by this exact solution, for the same case as in Fig. 4, except that $\left(n,l,m\right)=\left(1,5,3\right)$.